The relation between weight (mass) and force is something most people have a natural feel for: You have to apply more force to accelerate a heavy ball than a light one when throwing it. The formula is also straight forward:
F=m*acc

When you move in a curve, as when swinging a tennis racquet, you have something called swingweight that is supposed to tell you how it feels. The problem, however, is that most people don't know what swingweight really is. And even worse, the values you get from manufacturers and resellers are only valid when you rotate the racquet around a point 10 cm up the handle, a type of swing that rarely occurs in tennis.

I will try to shed some light on this and propose a curve where you can compare different racquets for different types of swings. Say that we have a racquet where we apply a force F at the handle and swing it around a point p. The the swing radius r says if the swing is short or long. When swinging we are interested in accelerating the racquet head:

If we look at the relation between F and the acceleration of the head acc we can define an equivalent mass me that tells us how much force you have to apply to get a certain acceleration, i.e. how heavy the racquets feels for different kinds of swings:
me=F/acc

We can then plot me and compare different racquets. But instead of plotting me against r I will plot it against 15/r. I that way we will get a convenient scale where 0 means moving it without rotation (a block) and 1 means whipping it around the wrist (5 cm outside the handle).

As you can see the Pro Staff is heavier for long swings (as expected), but it is also heavier for shorter swings despite that Cierzo has a higher swingweight. The reason is that the swingweight doesn't the the full story, even for a short swing. The diagram therefore gives you a way to compare these two racquets for different swings.

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For those who want hear some more details we need to define some lengths:

You can then calculate me in terms of the swing radius r:

Where m is the weight and sw is the swingweight of the racquet. I have used d=40 cm (i.e. 50 cm from the but) in the diagram above.

For those who want to play around with the figures I have an Excel-sheet that you can download here
Edit: There is an alternative and better version with a new excel sheet presented in post 40

Is there an ideal personal curve, you think, which will give a nice feel of the forces needed in the stroke.

That is a very good question!
No, I think that would be to push the significance of the curve a little to far. Different people will still like different racquet behavior. You will also vary the radius during the swing. But maybe you could take a racquet you like and use the curve as the basis when buying or customizing other racquets.

I would use the curve in the first post to say that the Pro Staff 90 is "always heavier, especially for long swings" something that is not obvious when you look at the weight, swingweight and balance.

I am having trouble filling in the formula. Could you provide an example of the calculations?

The excel sheet (see the first post) has two racquets included as an example.

Or do you mean how I derived the formula? I didn't include that since I didn't want to scare people with to much math to begin with. But it is coming, I just have to type it out so it looks nicer.

Quote:

And I don't get the 15/r part.

It is only there to rescale the curve to a more convenient form. It doesn't change the curve.

r is infinite when you move the racquet without rotation, but that is not so easy to show in a curve. 1/r = 0 when r is infinite, so thats easier to plot .

At the other end I picked 15 cm as a reasonable shortest swing radius. Since r is measured from 10 cm up the handle where the force is applied, r=15 means 5 cm outside the end but. Which is somewhere around the wrist. So when r=15 then 15/r=1. You can of course use another point as reference, r=10 or r=20 if you like, as long as you don't go all the way to r=0 when the calculation becomes irrelevant.

I don't mean how you got the formula figured out, but I am having trouble filling in the numbers and then get the correct me. I think I do it in the wrong units.
Let's say I have a racket with the following specs: 335g, 32cm balance, sw 330. Can you show me the math to calculate the me?

I don't mean how you got the formula figured out, but I am having trouble filling in the numbers and then get the correct me. I think I do it in the wrong units.
Let's say I have a racket with the following specs: 335g, 32cm balance, sw 330. Can you show me the math to calculate the me?

Your units looks fine.

In the Excel sheet the length c to the balance point (cg) is calculated from the HH/HL value. But if you want to enter the balance directly just type in your value -10 at c (in your case 22 cm). That will destroy the function that calculates it from HH/HL.

I don't know the length of your racquet, but assuming it is 69 cm the curve will look like this (using the new version of the spreadsheet):

I have also read Travlerajm's formulas and tried them. I don't know if you have heard of Mgr/I? If so, can this formula interact with yours?

It seems difficult, because according to Mgr/I, if I add weight at the top of the rackethandle, the relative speed of rackethead becomes higher relative to the hand. My virtual racket has a lower me at every point and the only outstanding difference between my racket and the prostaff in the example is that the prostaff has more weight. more static weight gives a higher Mgr/I, which should make it easier on the wrist. But then, according to your formula, the prostaff has more me at the wrist. Which should feel heavier then, I think. I am not questioning your formula, I just want to know if I may be comparing two different things.

On a side note. Once upon a time I tried adding weight on the top of the handle of a K Six-One 95 to speed up the racket head, but I never got it right. It kept feeling too heavy. Probably to much me.

I have experimented with your formula and I think it gives very valuable information. I am still experimenting with what spec change changes the curve in which way.

Could you share what combination of spec changes, makes the curve flatter/steeper, more curved or less curved?

It a challenge for me to figure out if you like a certain racket and thus the corresponding curve, if it acts like a blueprint.
Say you want the same feel during the stroke, but with just more me. Should the curve just being lifted upward or does the curve has to change a little to get the same feel? The trouble a lot players have, is that they like a certain racket, but want more heft. When adding weight it's tough to keep the same feel with a heavier racket.

I have experimented with your formula and I think it gives very valuable information. I am still experimenting with what spec change changes the curve in which way.

Could you share what combination of spec changes, makes the curve flatter/steeper, more curved or less curved?

It a challenge for me to figure out if you like a certain racket and thus the corresponding curve, if it acts like a blueprint.
Say you want the same feel during the stroke, but with just more me. Should the curve just being lifted upward or does the curve has to change a little to get the same feel? The trouble a lot players have, is that they like a certain racket, but want more heft. When adding weight it's tough to keep the same feel with a heavier racket.

I started this thread to get some input on how use the curve, so your view is a s good as mine

I would guess that a racquet that follows a similar curve but on a lower level (like your racquet compared to the Pro Staff) should feel similar, but a little lighter in all situations. A heavy, head light, low swingweight racquet will get a flatter curve than a more top heavy. What the curve can show is at what types of swings it actually is heavier, it might only be for very long swings, something that not is obvious when you just look at the basic data.

Let me give two examples: I have started from rather generic racquet, 70 cm long, weight 300 g, swingweight 320, even balance. I have then added 20 g of weight, in one case at the top, in the other at the handle. The latter only differs from the original at long swings and the curves quickly merge. The top heavy racquet is heavier all the way and difference increases for short swings:

In the other example I have also added 20 g, but in this case either all in the middle or half at the top and half at the handle. Putting everything in the middle increases the weight evenly, whereas the more "polarized" racquet increases a little more for short swings. But the difference is fairly small I would say.

Why not just use the parallel axis theorem to find the effective swingweights at the wrist, elbow, shoulder, etc.?

A good question. And the answer is that it is exactly what I am doing
The numerator in the equation in post 1 is Moment of Inertia around p

But choose to transform it into a weight for a couple of reasons:
- I think that weight is easier to understand
- The equivalent weight nicely converges to the weight of the racquet for very long swings, i.e. at the y-axis. Whereas the swingweight has no meaning when the racquet doesn't rotate.
- Weight gives the relation between force and acceleration whereas swingweight gives the relation between Moment and angular acceleration, again less intuitive.

I have this idea; I'll set-up 2 rackets, which I will try to get to feel the same, except that one is heavier overall. IOW. I want to achieve the same feel with both during play, but one will just feel heavier. I will try this with this method and if I succeed I will then carefully measure the specs.

I hope that you could make another plot of those two rackets. Then we can compare the different curves.

I know it's a personal experiment, but maybe we will learn something of it.

I have this idea; I'll set-up 2 rackets, which I will try to get to feel the same, except that one is heavier overall. IOW. I want to achieve the same feel with both during play, but one will just feel heavier. I will try this with this method and if I succeed I will then carefully measure the specs.

I hope that you could make another plot of those two rackets. Then we can compare the different curves.

I know it's a personal experiment, but maybe we will learn something of it.

It will be interesting to hear, just send me the data and I will plot them.